Hyperovals in Steiner triple systems

Quattrocchi, Gaetano; Zeitler, Herbert

doi:10.1007/bf01223811

Cited by 8 publications

(7 citation statements)

References 4 publications

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“…For Steiner systems that possess a maximal arc, we can therefore determine the file size. In addition, prior results in [32], [33], demonstrate that such maximal arcs exist in a large class of Steiner systems. In the discussion below, we make these arguments in a formal manner.…”

Section: A Fr Codes From Steiner Systemsmentioning

confidence: 87%

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Fractional Repetition Codes With Flexible Repair From Combinatorial Designs

Ölmez

Ramamoorthy

2016

IEEE Trans. Inform. Theory

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Fractional repetition (FR) codes are a class of regenerating codes for distributed storage systems with an exact (table-based) repair process that is also uncoded, i.e., upon failure, a node is regenerated by simply downloading packets from the surviving nodes. In our work, we present constructions of FR codes based on Steiner systems and resolvable combinatorial designs such as affine geometries, Hadamard designs and mutually orthogonal Latin squares. The failure resilience of our codes can be varied in a simple manner. We construct codes with normalized repair bandwidth (β) strictly larger than one; these cannot be obtained trivially from codes with β = 1. Furthermore, we present the Kronecker product technique for generating new codes from existing ones and elaborate on their properties. FR codes with locality are those where the repair degree is smaller than the number of nodes contacted for reconstructing the stored file. For these codes we establish a tradeoff between the local repair property and failure resilience and construct codes that meet this tradeoff. Much of prior work only provided lower bounds on the FR code rate. In our work, for most of our constructions we determine the code rate for certain parameter ranges.

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Section: A Fr Codes From Steiner Systemsmentioning

confidence: 87%

“…This demonstrates that file size calculations for Steiner systems cannot be performed just based on the system parameters. Accordingly, we consider Steiner systems that have maximal arcs [32], [33]. It turns out that we can determine the file size of the corresponding transposed codes.…”

Section: B Discussion Of Related Workmentioning

confidence: 99%

Fractional Repetition Codes With Flexible Repair From Combinatorial Designs

Ölmez

Ramamoorthy

2016

IEEE Trans. Inform. Theory

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“…Furthermore, there are no many general results regarding the existence of maximal arcs in Steiner systems. However, it is worth noting here that a Steiner system S(2, 3, θ) has at least one maximal arc if θ ≥ 7 and θ ≡ 3, 7 (mod 12) [26], and a Steiner system S(2, 4, θ) has a maximal arc of size θ+2 3 if θ ≥ 13, θ ≡ 1, 4 (mod 12), and θ−1 3 is a prime power [27]. Indeed, these two infinite families of Steiner systems that have maximal arcs can be of practical interest for real-world systems since the duals of designed FR codes have an applicable repetition degree of 3 or 4.…”

Section: B Optimal Constructions From Combinatorial Designsmentioning

confidence: 99%

On the Optimal Minimum Distance of Fractional Repetition Codes

Zhu

Shum

Wang

et al. 2020

Preprint

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Fractional repetition (FR) codes are a class of repair efficient erasure codes that can recover a failed storage node with both optimal repair bandwidth and complexity. In this paper, we study the minimum distance of FR codes, which is the smallest number of nodes whose failure leads to the unrecoverable loss of the stored file. We consider upper bounds on the minimum distance and present several families of explicit FR codes attaining these bounds. The optimal constructions are derived from regular graphs and combinatorial designs, respectively.

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“…In [6], de Resmini proved a general result which characterizes (n; 0, l − 1)-sets in S(2, l, v). Other results on hyperovals in Steiner systems can be found in [3,7].…”

Section: Introductionmentioning

confidence: 97%

Hyperovals in Steiner systems

Ling

2003

Journal of Geometry

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In this paper, we show that the basic necessary condition for the existence of a (k; 0, 2)-set in an S(2, 4, v) is also sufficient. It solves a problem posed by de Resmini [6] and we also prove some asymptotic results concerning the existence of hyperovals in Steiner systems with large block size. The results are generally applicable to designs with maximal arcs. (2000): 51E10, 05B05. Mathematics Subject Classification

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Hyperovals in Steiner triple systems

Cited by 8 publications

References 4 publications

Fractional Repetition Codes With Flexible Repair From Combinatorial Designs

Fractional Repetition Codes With Flexible Repair From Combinatorial Designs

On the Optimal Minimum Distance of Fractional Repetition Codes

Hyperovals in Steiner systems

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